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Is there a name for ring elements that multiply to zero?

Let $R$ be a ring (not necessarily with identity). Is there a name for an element $a\in R$ for which $ab=0$ for all $b\in R$? Similarly, for an element $a\in R$ for which $ba=0$ for all $b\in R$? Note, a unital ring has exactly one of each, just zero in both cases, as if $ab=0$ […]

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Logic uniqueness and injection

Definition of uniqueness is $\exists y [P(y) \land \forall x[P(x) \rightarrow x=y]]$. Then, $\exists y [P(y) \land \forall x[\lnot P(x) \lor x=y]]$. Distributing $P(y)$, we have $\exists y \forall x[P(y) \land \lnot P(x) \lor P(y) \land (x=y)]$. So, we have $\exists y \forall x[P(y) \subseteq P(x) \rightarrow P(y) \land (x=y)]$, then $\exists y \forall x[P(y) […]

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The density of for a couple of random variables

$X$ and $Y$ are two independent random variables distribute Uniform$[0,1]$, where $M = \min(X,Y)$, $N = \max(X,Y)$. I want to find the density of the couple $(M,N)$. We can start by finding the marginal distributions, doing so gives $$f_M(m) = 2(1-m),~~ m \in [0,1]$$ $$f_N(n) = 2n,~~ n \in [0,1] $$ I found this answer […]

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Prove that for every $c>0$ there exists an $a \in [0,1)$ with $f(a) > c$

Given a continuous function $f : [0,1) \rightarrow \mathbb{R}$, $f(0) = 0$, $f(x) >0$ for $x>0$ and $$ \lim_{x \rightarrow 1} \frac{1}{f(x)} = 0.$$ Prove that for every $c>0$ there exists an $a \in [0,1)$ with $f(a) > c$, also prove that $f([0,1)) = [0, \infty )$. Since $ \lim_{x \rightarrow 1} \frac{1}{f(x)} = 0$ […]

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density function and measure null

If $P = fd\mu $, $Q = gd\mu$ and $P = h dQ$ do we have $$ \mu(g=0)=0 $$ My goal is to show that $h = \frac{f}{g} $ almost $\mu$ everywhere. thanks and regards.

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it’s no use doing sth

Thank you for clicking into this page to help me. In the grammar book, I have learned that a gerund can be preceded by a pronoun (usually more acceptable, a possessive pronoun) to show who it is that does this action. And regarding gerund, there is a structure ‘it’s no use doing sth’ , where […]

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Puzzled by this heuristic argument for the prime number theorem

The following heuristic argument for the prime number theorem was taken from https://sites.williams.edu/Morgan/2008/10/11/heuristic-derivation-of-prime-number-theorem/. Frank Morgan attributes it to Hugh Bray via Greg Martin. Suppose that there is a nice probability function $P(x)$ that a large integer $x$ is prime. As $x$ increases by $\Delta x = 1$, the new potential divisor $x$ is prime with […]

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What is the probability that the lift safely reaches the ground when there are 10 adults in the lift?

The maximum weight that an elevator in an apartment complex can accommodate is 800kg. The average adult weight be about 70 kgs with a variance of 200. What is the probability that the lift safely reaches the ground when there are 10 adults in the lift?

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how do you change a 9 by 4 rectangle to a 6 by 6 square?

I know you have to do around the same thing as the 9 by 16 rectangle to a 12 by 12, but I need to know how to make my 9 by 4 into a 6 by 6 with 3 pieces. There are 2 ways. I need the other. If you can, can you please […]

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Number of possible integers less than 100000 such that the digits 4,5,6 appear in that order

I have been stuck on this problem for some time and I am not sure how to approach it: “How many positive integers less than 100,000 have digits containing 4,5,6 in that particular order?” I am thinking that I would multiply 10X10X3X2X1 since the two of the five digit spaces can hold any digit between […]